Brownian Sheet and Quasi-Sure Analysis

Details Brownian Sheet and Quasi-Sure Analysis Brownian Sheet and Quasi-Sure Analysis

2004-06-28

arxiv.org/abs/math/0406557v1

Mathematics

Probability

23 pages. Proceedings of the Fields Institute (to appear)

Scientific paper

We present a self-contained and modern survey of some existing quasi-sure results via the connection to the Brownian sheet. Among other things, we prove that quasi-every continuous function: (i) satisfies the local law of the iterated logarithm; (ii) has Levy's modulus of continuity for Brownian motion; (iii) is nowhere differentiable; and (iv) has a nontrivial quadratic variation. We also present a hint of how to extend (iii) to obtain a quasi-sure refinement of the M. Csorgo--P. Revesz modulus of continuity for almost every continuous function along the lines suggested by M. Fukushima.

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